New Research


2011/09/01

Statistics with Pseudo Random Numbers

Computer Intensive Statistics

Shingo Shirahata(Professor), Wataru Sakamoto(Associate Professor), Kei Hirose(Assistant Professor),

  In order to analyze statistical data, computer is a must-have item. However, the classical procedures are developed without the presupposition of the existing computers. Classical theory and methods are developed under the assumption that the underlying distributions follow normal distribution, binomial distribution, poisson distribution and their related distributions. This is because these distributions give beautiful theoretical results and reflect physical models, not because that they are convenient to use of computers. Especially, multivariate normal distribution is the unique one with beautiful theory. The improvement of the quality of Japanese manufactured goods after the Second World War is due to the statistical quality control under the assumptions of the above distributions.
  Recently situations where classical theory and models are not applicable are increasing. In these cases we need flexible model construction and procedures. Now the software and hardware of computer are rapidly developed. In such an atmosphere, B. Efron proposed Bootstrap method. His method consider the observations as the representative of the population and then samplings form the observations are performed by computer simulations. The procedure requires no concrete models and makes possible to test and interval estimate of the population parameters.
  The problem to estimate some parameter by computer simulation is already known (for example, Buffon’s needle problem). The sample figure is a simple trial to estimate the value pi. Here many pairs of uniform random numbers in unit interval are generated and count the number of pairs occurred in the fan shaped area. This is a very simple problem. Nowadays many statistical procedures are developed for more complex problems by using computer simulations. The methods to generate Pseudo random numbers are also developed (Markov chain Monte Carlo method, Mersenne twister method and so on).
  Friedman et al. proposed CART method which is a representative nonlinear statistical method where visualization of data and iterative calculations are performed. Many hybrid procedures with high efficiencies are also proposed where simulation and iteration are combined. Statistical procedures using pseudo random numbers and iterations are, now, at the main stream of Statistics.
  The members of our study group including students in Ph.D. course having full-time job do researches computer intensive statistical procedures on, for example, Bayes statistics, smoothing procedures, model selections, construction of experimental designs, robust procedures, statistical graphics, all of which are based on computer simulations and iterative methods.


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